When people speak of “the amount of charge” in a battery or
capacitor, usually it is a misnomer, of the kind that leads to serious
misconceptions. It would be better to call it the gorge, and
to speak of gorging and disgorging the device. Charge is something else.

In particular, charge is strictly conserved, but gorge is not. A real
battery has some finite shelf life, and a real capacitor has some
nonzero internal leakage. The battery just sitting there will
disgorge itself. It would be a tremendous mistake to think that
“charge” is being lost, because that would violate conservation of
charge.

Consider the situation shown in part A of figure 1.
We have a sphere that carries a charge of +4 units on the upper
hemisphere and another +4 units on the lower hemisphere. Presumably
there is – somewhere – a counter-charge of −8 units, but we assume
it is so far away that the details don’t matter. The sphere has been
cut along the equatorial plane, so that that there is a tiny gap of
size g between the upper hemisphere and the lower hemisphere.
However, for present purposes the gap doesn’t matter very much.
Basically what we have is a sphere with a total charge of +8 units.

We now turn to part B of figure 1. As before,
there is a charge of +4 units on the upper hemisphere ... but now
there is zero charge on the lower hemisphere. We assume the gap is so
small that the mutual capacitance between the hemispheres is enormous
compared to the self-capacitance. We can figure out (by symmetry or
otherwise) that the 4 units of charge will distribute themselves
evenly over the sphere. In the far field, the field exhibits (to
leading order) a monopole field pattern, which is what we would expect
from +4 units of charge. There will also be a dipole term, due to
the voltage difference between the two hemispheres, but in the far
field, compared to the monopole term, the dipole term will be
negligible. We can understand this as follows: There will be a
significant field in the gap, but because the gap is small the mutual
capacitance is large, so the voltage difference between the
hemispheres will be small.

Finally, we turn to part C of figure 1. As before,
there is a charge of +4 units on the upper hemisphere ... but now
the charge on the lower hemisphere is −4 units. In the far field,
this looks like a dipole. It won’t be a very strong dipole, because
the separation between the positive and negative charges is not large,
but in the far field this small dipole term is all there is.

Now we get to the tricky part. Consider the following contrast:

Suppose we say object C is a sphere, and we ask how much
charge is on the sphere. The answer, obviously, is zero charge.
To be more specific, you could talk about the total charge or
the net charge or the honest-to-goodness charge ... but it seems
simpler and better to just call it charge.

Suppose we say object C is a capacitor, and we ask how much
«charge» is on the capacitor. The answer, according to
long-established convention, is that the capacitor has +4 units
of «charge». I write «charge» in scare quotes here,
because we have a problem.

The problem is, the word “charge” is being used in two inconsistent
ways. To solve this problem, let’s define the gorge, G, so
that

Note that according to the laws of physics, the charge Q is
conserved. Specifically, Q obeys a strict local conservation
law. In contrast, gorge is not conserved. Not at all. If you put
many units of gorge on a capacitor – or a battery – and come back
later, you might find that all of the gorge has disappeared, due to
some internal leakage process.

Note that what we have here is just a special case of the
three-terminal capacitor: The upper hemisphere, the lower hemisphere,
and the far-away counter-electrode. The terminology of charge and
gorge is not particularly good for handling three-terminal capacitors
in general ... but the case we have considered here is so exceedingly
common that it is worth having special tools for handling it.

The concept applies equally well to batteries as well as capacitors.
In real-world situation, the total charge Q might be 10 or 12 orders
of magnitude smaller than the gorge G. In such a situation the
total charge Q is sometimes called “stray” charge. As another way
of saying more-or-less the same thing: In normal operation, to an
excellent approximation, you have have a current flowing “into” a
capacitor; instead you have a current flowing through a
capacitor, i.e. in one terminal and out the other.